Influence of Bottom Topography on Long Water Waves

Details Influence of Bottom Topography on Long Water Waves Influence of Bottom Topography on Long Water Waves

2007-01-08

arxiv.org/abs/math/0701227v1

Mathematical Modelling and Numerical Analysis 41, 4 (2007) 771-799

Mathematics

Analysis of PDEs

Scientific paper

10.1051/m2an:2007041

We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotics expansion of the involved Dirichlet-Neumann operator. then, following the global strategy introduced by Bona, Colin and Lannes, new symetric asymptotic models are derived for each regime of bottom topography. Solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.

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